We introduce a new regression framework designed to deal with large-scale, complex data that lies around a low-dimensional manifold. Our approach first constructs a graph representation, referred to as the skeleton, to capture the underlying geometric structure. We then define metrics on the skeleton graph and apply nonparametric regression techniques, along with feature transformations based on the graph, to estimate the regression function. In addition to the included nonparametric methods, we also discuss the limitations of some nonparametric regressors with respect to the general metric space such as the skeleton graph. The proposed regression framework allows us to bypass the curse of dimensionality and provides additional advantages that it can handle the union of multiple manifolds and is robust to additive noise and noisy observations. We provide statistical guarantees for the proposed method and demonstrate its effectiveness through simulations and real data examples.

翻译：我们引入了一个新的回归框架，旨在处理围绕低维流形的大规模、复杂数据。我们的方法首先构建了一个图形表示，称为骨骼（skeleton），以捕获潜在的几何结构。然后，我们定义了基于该骨骼图形的度量，并应用了基于图形的特征转换和非参数回归技术来估计回归函数。除了包括的非参数方法外，我们还讨论了某些非参数回归估计器在骨骼图形等通用度量空间中的限制。所提出的回归框架使我们能够摆脱维度灾难，并提供了额外的优势，即能够处理多个流形的并集，并且对于加性噪声和嘈杂观测具有鲁棒性。我们为所提出的方法提供了统计保证，并通过模拟和实际数据示例证明了其有效性。